Exercise 4, § 4

Let G be a locally compact group. For every $t \in \mathbf{R}_+^*$, let $\mu_t$ be a nonzero bounded positive measure on G. Assume that the mapping $t \mapsto \mu_t$ is continuous for the topology $\sigma(\mathscr{M}^1(G), \overline{\mathscr{K}(G)})$, and that $\mu_{s+t} = \mu_s * \mu_t$ (s, t in $\mathbf{R}_+^*$). a) Show that there exists a number $c \in \mathbf{R}$ such that $\| \mu_t \| = \exp(ct)$. (Observe that $t \mapsto \| \mu_t \|$ is lower semi-continuous and that $\| \mu_{s+t} \| = \| \mu_s \| \cdot \| \mu_t \|$. Make use of Prop. 18.) b) Suppose that $c = 0$. Show that $\mu_t$ converges weakly as $t \to 0$ to the normalized Haar measure of a compact subgroup of G. (Make use of the weak compactness of the unit ball of $\mathscr{M}^1(G)$, and show that for every weak cluster point $\mu$ of $t \mapsto \mu_t$ with respect to the filter of neighborhoods of 0 in $\mathbf{R}_+^*$, one has $\mu_t * \mu = \mu_t$ for every $t$, then $\mu * \mu = \mu$; next, apply Exer. 3.)